91Ó°ÊÓ

The surface generated by revolving the circle \(r=\cos \theta\) about the line \(\theta=\pi / 2\)

Find the exact arc length of the curve over the stated interval. $$ x=\sqrt{t}-2, y=2 t^{3 / 4} \quad(1 \leq t \leq 16) $$

List the forms for standard equations of parabolas, ellipses, and hyperbolas, and write a summary of techniques for sketching conic sections from their standard equations.

Find the exact arc length of the curve over the stated interval. $$ x=\cos 3 t, y=\sin 3 t \quad(0 \leq t \leq \pi) $$

Find the exact arc length of the curve over the stated interval. $$ x=\sin t+\cos t, \quad y=\sin t-\cos t \quad(0 \leq t \leq \pi) $$

(a) Show that if \(A\) and \(B\) are not both zero, then the graph of the polar equation $$ r=A \sin \theta+B \cos \theta $$ is a circle. Find its radius. (b) Derive Formulas (4) and (5) from the formula given in part (a).

The sphere of radius \(a\) generated by revolving the semicircle \(r=a\) in the upper half-plane about the polar axis.

Find the exact arc length of the curve over the stated interval. $$ x=e^{2 t}(\sin t+\cos t), y=e^{2 t}(\sin t-\cos t)(-1 \leq t \leq 1) $$

Find the highest point on the cardioid \(r=1+\cos \theta\).

Writing In order to find the area of a region bounded by two polar curves it is often necessary to determine their points of intersection. Give an example to illustrate that the points of intersection of curves \(r=f(\theta)\) and \(r=g(\theta)\) may not coincide with solutions to \(f(\theta)=g(\theta) .\) Discuss some strategies for determining intersection points of polar curves and provide examples to illustrate your strategies.